In the Anglo-Saxon epic poem, Beowulf, the protagonist Beowulf insists on battling Grendel with his bare hands, instead of using a weapon, because, he says, Grendel doesn't use one.

The sense of honor Beowulf displays here is one of the main themes of the poem. Beowulf sees using a weapon against a beast that uses no weapon as dishonorable. He will fight Grendel on equal terms. It is the honorable thing to do. Grendel is charmed and cannot be cut by swords, but Beowulf does not know this as far as a reader can tell, and this is not why, according to his own words, he refuses to use a sword.

As he says:

...I have heard,

Too, that the monster's scorn of men

Is so great that he needs no weapons and fears none.

Nor will I. My lord Higlac

Might think less of me if I let my sword

Go where my feet were afraid to, if I hid

Behind some broad linden shield: my hands

Alone shall fight for me, struggle for life

Against the monster. God must decide

Who will be given to death's cold grip. (261-270)

Of course, no one else mentions this, and others have fought Grendel with swords. But this is one of the character traits that makes Beowulf different and special. His sense of honor is even greater than that of the rest of his society. That's why he's an epic hero. He could also give a great speech!

Beowulf is in search of glory and he hears of the terrible mess Herot and Hrothgar are in and is determined to make a name for himself. He states that he will fight this monster without weapons, and there are several reasons for his decision. First, Beowulf states that Grendel does not use weapons, and Beowulf wants to fight in the same way. Another, more personal reason he mentions is that his king, Higlac, would be disappointed in him if he were to fight this monster in any other way. Beowulf is a mighty warrior and knows his own strength. He has boasted about ridding the earth of giants and the seas of monsters, so he does not feel the need to fight with weapons.

I am not sure how Beowulf knows this, but the reason that he does not use weapons when fighting Grendel is that the monster has some sort of magical protection. The monster has managed to do something that prevents any sort of edged weapon from doing him harm.

So, instead of trying to kill the monster with a sword or spear or anything like that, Beowulf fights him hand to hand and tears his arm off.

You can find this in Chapter 12 if you need to be able to cite proof of this answer.

Beowulf is a mighty warrior of epic proportions and he must reprove himself each time. He is always up for a challenge. He is aware that Grendel is no ordinary beast and that he is worthy of a fair opponent. He is also aware that he exists as some magical creature who has powers beyond man's strength. When Beowulf goes to fight Grendel he wants to be at his own best advantage. He feels that a fair fight with no weapons would be the best way.

By not using weapons Beowulf even furthers the decision of the people that he is a true hero.